What Is the Geometric Difference in Intersection Forms Q(a,b) vs Q(b,a)?

[WWGD]

Hi, everyone:

This should be easy, but I am having trouble with it. I am rusty and trying

to get back in the game:

Let Q(a,b) be an intersection form in the middle homology class

of some 2n-manifold.

What is the geometric difference between Q(a,b) and Q(b,a).?

If n is even, they are of the same sign, opposite sign

if n odd, but I am not clear on what the geometric

difference is with the different orders.


2) Also: Am I missing something really obvious here:

If H_n==0 for a 2n-manifold M . Does it follow (by bilinearity)

that Q==0.?. Since the only class is the zero class, it

would seem to follow right away. What is the geometry behind

this.?. I understand that this does not imply that there is

no actual intersection, but that the (signed) net intersection

is zero. (If above is correct) Anyone have an insight on the

geometry behind this.?

Thanks.

 

[pat_connell]

For the first question, the geometric difference between Q(a,b) and Q(b,a) is that the sign of the intersection form changes depending on whether n is even or odd. For an even n, the sign is the same for both, while for an odd n the sign is opposite. For the second question, if H_n == 0 for a 2n-manifold M, then it does follow that Q==0 by bilinearity. The geometry behind this is that since the only class in the middle homology is the zero class, the intersection form over the zero class is zero. This does not imply that there is no actual intersection, as you mentioned, but that the net intersection is zero.